Recently, mMy final exam is getting nearer and I am feeling very anxious and nervous, overthinking twhat whatould happen if I do not get a good result.

(1) "Recently" is redundant here, since "getting nearer" already implies it.
(2) The phrase is "overthink (something)".

I feelt stressed out withby the thought that I might disappoint my parent(s) if I do not meet their expectations.

(1) "Feel" would be more expected here, since in the previous sentence you also used the past tense ("I am feeling very anxious and nervous").
(2) The more natural expression would be "stressed out by (something)".
(3) "Expectations" would sound more natural, since you don't specify what that expectation is.

I also seek help from my tutor who advices me to do some meditation.

"Tutor" is singular.

All these techniques help to reduce my academic pressure but do not help much with the increasing anxiety.

But one thing I've noticed about myself these days is that whenever I feelt overwhelmed, I am apt to eat more than usual.

(1) "I noticed" and "these days" don't go together. This is because "these days", specifically "these", implies that whatever action or situation you're referring to is still continuing up to the present, in which case "have noticed" is expected. If you insist on using "I noticed", you can consider: "one thing I noticed about myself in the past few days".
(2) To keep the tense consistent, it should be "feel" and not "felt", since "I am apt to eat more than usual" is not in the past tense.

Normally, I would skip dinner as a way of maintaining my weight , but recently, I feel much better after having dinner, and study with a better concentration.

Generally, "concentration" in this context is uncountable. That being said, "with a better concentration" might be okay; to me, it just sounds more literary, and not necessarily wrong.

# Can you tell me alternative words to use instead of "these days" and "recently"?

Below are some examples of alternatives in practice, based on the sentences you've written. Also, sometimes, you can even use completely different phrases that aren't necessarily synonyms, but which are specific to the context.

"But one thing I've noticed about myself AS OF LATE is that..."
"But one thing I've noticed about myself LATTERLY is that..."
"But one thing I've noticed about myself, ADMIST THIS HECTIC PERIOD, is that..." (Here, "admist this hectic period" is understood to be referring to the present, since you've been describing it as hectic.)
"Normally, I would skip dinner as a way of maintaining my weight, but NOWADAYS, I feel much better..."
"Normally, I would skip dinner as a way of maintaining my weight, but LATELY, I feel much better..."

# Can you comment on some awkward sentence transitions and how I should fix them?

Insofar as sentence transitions are concerned, I think you did a good job! Your writing flows quite nicely.

Recently, mMy final exam is gettoing nearerto take place soon and I am feeling very anxious and nervous, overthinking that whatabout the consequences if I do non't get a good result.

I felt stressed out with the thought that I might disappoint my parent(s) if I do not meet their expectation(s).

To calm myself down, I draw and write often, trying to distract my mind from negative thoughts.

I also seek help from my tutor who advices me to do some meditation.

All these techniques help (me) to reduce my academic pressure but do not help much with the increasing anxiety.

But one thing I noticed about myself these (past) days is that whenever I felt overwhelmed, I am apt to eat more than usual.

Normally, I would skip dinner as a way of maintaining my weight ,but recently, I feel much better after having dinner, and study with a bettermore concentration.

# Can you tell me alternative words to use instead of "these days" and "recently"?

"Currently", "in the moment", "right now" can be used in place of "recently" and "these days". You can even say "since the past few days" or "since the last few days".

# Can you comment on some awkward sentence transitions and how I should fix them?

I don't think there are any awkward sentence transitions.